Let C be the solid cone with the boundary surfaces x^{2}+y^{2}=z^{2} and z=0. The density of the solid at point (x,y,z) is z.

Gisselle Hodges

Gisselle Hodges

Answered question

2022-10-29

Finding volume of a cone given density
Let C be the solid cone with the boundary surfaces x 2 + y 2 = z 2 and z = 0. The density of the solid at point (x,y,z) is z.
Find the volume of the solid using the integrals in both the cylindrical coordinates and the spherical coordinates.
I really cant do this question. I know that V = ρ / m but what is m?
I also don't understand how you can even find the volume with integrals because there is no limit of z given so it would just be infinity wouldn't it.
Can someone start me off, I obviously don't need anyone to show me how to integrate. Just struggling on what the density and stuff is relevant.

Answer & Explanation

Szulikto

Szulikto

Beginner2022-10-30Added 22 answers

Explanation:
As a start cylindrical coordinates case is simpler perhaps by cylinders stacking summation d V = ρ π r 2 d z
ρ = z , r = z
V = π z 3 d z = z 4 π / 4 etc...

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